Instantaneous rate of change of a sphere

In summary, the instantaneous rate of change of a sphere is given by the formula r(t) = 4πr^2, where r is the radius of the sphere and t is the time. It can be calculated by taking the derivative of the formula with respect to t. The instantaneous rate of change represents the rate at which the surface area of the sphere is changing at a specific moment in time, and is directly related to its volume. It has many real-world applications in physics, engineering, and mathematics, including calculating heat transfer, fluid flow rate, and object speed, as well as solving optimization problems.
  • #1
ivysmerlin
2
0

Homework Statement


Find the instantaneous rate of change of V with respect to r when r=5 micrometers


Homework Equations



V=4/3pi(r^3)

The Attempt at a Solution



would you just take the derivative? and if so, wouldn't it just be zero, because it comes out to be a real number, right? but that doesn't seem right...
 
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  • #2
What is the derivative of V=4/3pi(r^3)? Now what would the value be at r=5micrometers? Not zero.
 
  • #3
ooh, I am an idiot, thanks
 

1. What is the formula for instantaneous rate of change of a sphere?

The formula for instantaneous rate of change of a sphere is given by r(t) = 4πr^2, where r is the radius of the sphere and t is the time.

2. How is instantaneous rate of change of a sphere calculated?

Instantaneous rate of change of a sphere is calculated by taking the derivative of the formula r(t) = 4πr^2 with respect to t. This will give us the instantaneous rate of change at any given time t.

3. What does the instantaneous rate of change of a sphere represent?

The instantaneous rate of change of a sphere represents the rate at which the surface area of the sphere is changing at a specific moment in time. It is also known as the instantaneous rate of change of surface area.

4. How does the instantaneous rate of change of a sphere relate to its volume?

The instantaneous rate of change of a sphere is directly related to its volume. This is because the volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere. Taking the derivative of this formula will give us the instantaneous rate of change of volume, which is dV/dt = 4πr^2(dr/dt). As we can see, the instantaneous rate of change of volume is directly proportional to the instantaneous rate of change of surface area (which is dr/dt).

5. How can the instantaneous rate of change of a sphere be used in real-world applications?

The instantaneous rate of change of a sphere has many practical applications in fields such as physics, engineering, and mathematics. For example, it can be used to calculate the rate of heat transfer in a spherical object, the flow rate of a fluid through a spherical container, or the speed of a rolling ball. It is also used in optimization problems, where we want to find the maximum or minimum value of a function (such as volume or surface area) at a specific point in time.

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